Yu.P. Kunashenko1,2
1Tomsk Polytechnic University2Tomsk State Pedagogical University
CoherentBremsstrahlung from
planar channeled positron
«RREPS-13» and ''Meghri-13''
Coherent bremsstrahlung and coherent electron-positron pair photoproduction
was predicted by
Ter-Mikaelian High Energy Electromagnetic processes in Condensed Media, Wiley Interscience, New-York, 1972.
Uberall H. High-energy interference effect of bremsstrahlung and pair production in crystals. // Physical review volume 103, number 4. 1956. p.1055-1067.
Now coherent effects are divided into two types: A and B Saenz A.W. and Uberall H., Theory of coherent bremsstrahlung. -
in: Coherent Radiation Sourses, eds. A.W. Saenz and H. Uberall. - Berlin: Springer-Verlag.- 1985.- p.5-32.]
«RREPS-13» and ''Meghri-13''
Review of experimental and theoretical work on coherent type
A bremsstrahlung and coherent electron-positron pair photoproduction in crystals was given by Diambrini - Palazzi G . // Rev.Mod.Phys., 1968, v.10, p.611.
Combined effect arise when channeled particle emits coherent bremsstrahlung and manifest it self in changing of coherent peaks
For the first time combined effect in coherent type B bremsstrahlung was observed in Tomsk
Amosov K.Yu., Vnukov I.E., Naumtnko G.F., Potulisin A.P., Saruchev V.P. Pis’ma v ZhETP 1992 v. 55 p. 587
.
«RREPS-13» and ''Meghri-13''
Discreet axis
Planar continuous potential
Positron moves through crystal in the regime of planar channeling.Longitudinal positron momentum is parallel to crystal axis.
«RREPS-13» and ''Meghri-13''
Photon scattering by channeled positron
;2
,12
3
3
||32
k
pd
dddJ
EEMdffififi
n ni
nifn
ni
nifnfi EEEE
M
ReAeAeRe 21
1
12
We use virtual photon methods therefore we need cross-section of Photon scattering by channeled positron. It is convenient to consider scattering of photons by channeled positron in a coordinate system moving with the longitudinal velositi of channeled positron. In the moving reference frame one can use non relativistic approach to the scattering process.
Cross - section
matrix element (n is intermediate state)
J initial positron flux
«RREPS-13» and ''Meghri-13''
Photon scattering by channeled positron
rrrkprR 2 dich
mc
elkkl exp
2ˆ
2
22
rrrkprA 1 dich
mc
elkkl exp
2ˆ
1
22
matrix element of the photon absorption
matrix element of photon emission
«RREPS-13» and ''Meghri-13''
krr ich
ph exp2 22
||||exp rkr fififi ix
02
1 2
xxeVp
m fixfi 2
204
xd
VxeV
xHxCx nnn
2exp
22
!24/1
2/1
nC
nn
md
Vm 022
,
2
1nn
Photon wave function
Channeled positron wave function
Channeled positron transverse wave function
Photon scattering by channeled positron
«RREPS-13» and ''Meghri-13''
.
2
2
11
2
21
1
2
1
2
22
22
ddd
EE
EEEEcmd
f
fifi
nni
nifn
ni
nifn
||
||||||||
1
k
kkkk
ReAeAeRe
Photon scattering by channeled positron
0
21
1
i
fi
fi
EE
||
||||||||
k
kkkk
From the δ function it is follows conservation laws:
In the initial state i positron at rest
«RREPS-13» and ''Meghri-13''
m
kkc
k
ck
c
yfzffi
yf
zf
2
sin0
cos
2,||
2,||
21
,||
2,||
1
Photon scattering by channeled positron
If the incident photon moves along the OZ axis then its wave vector is defined as follows:
ckk /,,0,0 k
the wave vector of a scattered photon
ckkkk /,cos,sinsin,cossin k
2
42
2
111
22
12
2
12
2
sin42cos41
cos
ctgcm
mcmcmc
mc
fi
Photon scattering by channeled positron
Connected between scattered and initial photon energies
fi 12
(“(“Compton Compton formulaformula”)”)
d
dfi
«RREPS-13» and ''Meghri-13''
«RREPS-13» and ''Meghri-13''
The cross section of the coherent bremsstrahlung from channeled positron
.exp)(,|)(|)( 11
1
R
r
r
ZerVrrVrV
N
ii
electrostatic potential of the crystal axis
dSgku
cB
du
cBL
NeZ
dn
ngn
2
1
22
2
22
exp2
exp
)(
The spectrum of virtual photons of crystal axis
1,
/ln
22
2
aRc
aL
22
2
,1)()1()( uRc
xxEiexxB x
Here d is the lattice constant, ω is the frequency of virtual photon
/mc is Compton wave length
2u is the mean-square displacement of a crystal atom from equilibrium position
R is the screening radius, N is the number of atoms in the axis, Z is atomic number.
«RREPS-13» and ''Meghri-13''
The cross section of the coherent bremsstrahlung from channeled positron
If we neglect the terms c in the functions L and B
than we obtain the virtual photons spectrum from well – known
Ter- Mikaelian book.
dngn
/2
Ei(-x) is the exponential integral function,
is the 1D reciprocal lattice vector,
S is the structure factor of a crystal axis, γ is the relativistic factor.
«RREPS-13» and ''Meghri-13''
ddr
rfi,
The cross section of the coherent bremsstrahlung from channeled positron
For transform the photons scattering cross-section into the laboratory coordinate system we use the “Compton “Compton formula”formula” and the Lorentz transformation for the
photon energy (the Doppler shift): cos1 r
is the energy of emitted photon
is the energy of virtual photon
r
«RREPS-13» and ''Meghri-13''
The cross section of the coherent bremsstrahlung from channeled positron
MAX
MIN
dnddd
r
rfi
r
r
CR
fi
)(,
The limits of integration are determined by the condition 1cos1
1
fir
MIN
1
1fir
MAX
Results of calculationResults of calculation
fir
r
CR
fi
d
r
rTM
d
0 .0 0 0 .0 1 0 .0 2 0 .0 3 0 .0 4 0 .0 5 0 .0 6 0 .0 70
5 . 1 0 2 8
1 . 1 0 2 7
1 .5 1 0 2 7
2 . 1 0 2 7
2 .5 1 0 2 7
, M e V
st
,cm
2
W (100) planeW (220) axisNs=104
2
«RREPS-13» and ''Meghri-13''
0 .0 0 0 .0 1 0 .0 2 0 .0 3 0 .0 4 0 .0 5 0 .0 6 0 .0 7
5 . 1 0 2 8
1 . 1 0 2 7
1 .5 1 0 2 7
2 . 1 0 2 7
2 .5 1 0 2 7
3 . 1 0 2 7
3 .5 1 0 2 7
, M e V
,
cm2
fir
r
CR
fi
r
rTM
dNd
1
Results of calculationResults of calculation
«RREPS-13» and ''Meghri-13''
The cross section of the coherent bremsstrahlung from channeled positron
Conclusion
It is possible the coherent bremsstrahlung from planar channeled positron
The contribution to cross-section of the coherent bremsstrahlung from channeled positron result in splitting of coherent peek and appearance of fine structure of photon spectrum